aula 4Função de densidade condicional

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7.3 Conditional probability density function
We often wish to use observations to indirectly learn about other unobserved quantities. In order to do so, we
must define how the observation of a random variable changes the distribution of another random variable. This
is formally done by using the conditional probability density function, presented in Definition 7.20
Definition 7.20. Let X and Y be two vectors of random variables. The pdf of X given Y is defined as
fX|Y(x|y) =
f(X,Y)(x,y)
fY(y)
The conditional pdf, fX|Y(x|y), is the density function of X given that Y = y. When (X,y) is discrete, there
exists a straightforward justification for this definition. Indeed,
Lemma 7.21. If (X,Y) is discrete, then
fX|Y(x|y) = P (X = x|Y = y)
Proof.
fX|Y(x|y) =
f(X,Y)(x,y)
fY(y)
Definition 7.20
=
P (X = x,Y = y)
P (Y = y)
Definition 7.6
= P (X = x|Y = y) Definition 2.42
When (X,Y) is continuous, the justification is more technical. This occurs because P (Y = Y) = 0 and,
therefore, one cannot directly use Definition 2.42 to condition on {Y = y}. However, informally, if dx ≈ 0 and
dy ≈ 0, then
fX|Y(x|y)dx =
f(X,Y)(x,y)dxdy
fY(y)dy
Definition 7.20
≈ P (x ≤ X ≤ x + dx,y ≤ Y ≤ y + dy)
P (y ≤ Y ≤ y + dy) Definition 7.6
= P (x ≤ X ≤ x + dx|y ≤ Y ≤ y + dy) Definition 2.42
In words, if (X,Y) is continuous, then fX|Y(x|y) determines how much probability is concentrated near x given
that Y = y.
Example 7.22. In Example 7.10,
fX|Y (0|0) =
f(X,Y )(x, y)
fY (y)
=
0.2
0.5
= 0.4 Definition 7.20
fX|Y (0|1) =
0.4
0.5
= 0.8
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Also observe that
fX|Y (1|0) = P (X = 1|Y = 0) Lemma 7.21
= 1− P (X = 0|Y = 0) Lemma 2.4
= 1− fX|Y (0|0) = 0.6 Lemma 7.21
fX|Y (1|1) = 1− fX|Y (0|1) = 0.2
Example 7.23. Let (X,Y ) be a continuous vector of random variables such that
f(X,Y )(x, y) =
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2
x(2− x− y)I(x > 0)I(y < 1)
Notice that
fY (y) =
∫ ∞
−∞
f(X,Y )(x, y)dx Lemma 7.11
=
∫ 1
0
15
2
x(2− x− y)I(y < 1)dx
=
15
2
(
2
3
− y
2
)I(y < 1) (18)
Therefore,
fX|Y (x|y) =
f(X,Y )(x, y)
fY (y)
Definition 7.20
=
15
2 x(2− x− y)I(x > 0)I(y < 1)
15
2 (
2
3 − y2 )I(y < 1)
eq. (18)
=
x(2− x− y)
2
3 − y2
I(x > 0)
=
6x(2− x− y)
4− 3y I(x > 0)
Definition 7.24.
P (X ∈ A|Y = y) =
∫
A
fX|Y(x|y)dx
Conditional pdf’s have similar properties as marginal pdf’s. For example, if one integrates out one of the
coordinates of a conditional pdf, then one obtains the conditional pdf of the remaining coordinates. This result
can be seen as a generalization of Theorem 2.66 to random variables. The formal result is stated in Theorem 7.25.
Theorem 7.25 (Law of total probability for vectors of random variables).∫ ∞
−∞
fX|Y(x|y)dxi = fX−i|Y (x−i|y)
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Proof. ∫ ∞
−∞
fX|Y(x|y)dxi =
∫ ∞
−∞
f(X,Y)(x,y)
fY(y)
dxi Definition 7.20
=
1
fY(y)
∫ ∞
−∞
f(X,Y)(x,y)dxi
=
1
fY(y)
f(X−i,Y )(x−i,y) Lemma 7.11
= fX−i|Y (x−i|y) Definition 7.20
Also, the conditional distribution of X given Y can be obtained from the distribution of Y given X and the
marginal distribution of X. This result is presented in Theorem 7.26 and is a generalization of Bayes Theorem
for vectors of random variables.
Theorem 7.26 (Bayes theorem for vectors of random variables).
fX|Y(x|y) =
fX(x)fY|X(y|x)∫∞
−∞ fX(x)fY|X(y|x)dx
Proof.
fX|Y(x|y) =
f(X,Y)(x,Y)
fY(Y)
Definition 7.20
=
fX(x)fY|X(y|x)
fY(Y)
Definition 7.20
=
fX(x)fY|X(y|x)∫∞
∞ f(X,Y)(x,y)dx
Lemma 7.11
=
fX(x)fY|X(y|x)∫∞
−∞ fX(x)fY|X(y|x)dx
Definition 7.20
Example 7.27. Let X ∼ Gamma(a, b). Also, Y |X = x ∼ Exponential(x). We can use Theorem 7.26 to find the
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distribution of X given Y . Note that
fX|Y (x|y) =
fX(x)fY |X(y|x)∫∞
−∞ fX(x)fY |X(y|x)dx
=
ba
Γ(a)x
a−1 exp(−bx) · x exp(−xy)∫∞
−∞
ba
Γ(a)x
a−1 exp(−bx) · x exp(−xy)dx
=
xa exp(−(b+ y)x)∫∞
−∞ x
(a+1)−1 exp(−(b+ y)x)dx
=
xa exp(−(b+ y)x)
Γ(a+1)
ba+1
∫∞
−∞
ba+1
Γ(a+1)x
(a+1)−1 exp(−(b+ y)x)dx
=
xa exp(−(b+ y)x)
Γ(a+1)
ba+1
· 1
Definition 5.41
=
ba+1
Γ(a+ 1)
xa exp(−(b+ y)x)
That is, X|Y = y ∼ Gamma(a+ 1, b+ y).
7.3.1 Independence
Independence describes a particular type of conditional density that is commonly used in Statistics. Informally, X1
and X2 are independent if, no matter what value is observed for X1, this observation brings no information about
X2 (and vice-versa). This is a generalization of the concept of independence between events (Definition 2.46).
Independence between random vectors is formally presented in 7.28
Definition 7.28. We say that X1, . . . ,Xd are conditionally independent given Y if, for every x1, . . . ,xd and y,
f(X1,...,Xd)|Y(x1, . . . ,xd|y) =
d∏
i=1
fXi|Y (xi|y)
In particular, we say that X1, . . . ,Xd are independent if Y is empty, that is, for every x1, . . . ,xd
f(X1,...,Xd)(x1, . . . ,xd) =
d∏
i=1
fXi(xi)
Example 7.29. Consider that
fX1,X2|θ(x1, x2|t) = tx1+x2(1− t)2−x1−x2I(x1 ∈ {0, 1})I(x2 ∈ {0, 1})
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